Opportunistic Spectrum Access in Multiple Primary User Environments Under the Packet Collision Constraint

Transcription

1 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. X, NO. Y, 1 Oortunistic Sectrum Access in Multile Primary User Environments Under the Packet Collision Constraint Eric Jung, Student Member, IEEE, and Xin Liu, Member, IEEE Abstract Cognitive Radio (CR) technology has great otential to alleviate sectrum scarcity in wireless communications. It allows secondary users (SUs) to oortunistically access sectrum licensed by rimary users (PUs) while rotecting PU activity. The rotection of the PUs is central to the adotion of this technology, since no PU would accommodate SU access to its own detriment. In this aer, we consider an SU that must rotect multile PUs simultaneously. We focus on the PU acket collision robability as the rotection metric. The PUs are unslotted and may have different idle/busy time distributions and rotection requirements. Under general idle time distributions, we determine the form of the SU otimal access olicy, and identify two secial cases for which the comutation of the otimal olicy is significantly reduced. We also resent a simle algorithm to determine these olicies using rinciles of convex otimization theory. We then derive the otimal olicy for the same system when a SU has extra side information on PU activity. We evaluate the erformance of these olicies through simulation. Index Terms Cognitive Radio, Dynamic Sectrum Access, Sectrum Sharing. I. INTRODUCTION The static nature of sectrum regulatory olicy in the U.S. has led to an artificial scarcity of available sectrum. The FCC has estimated that 6% of the sectrum below 6 GHz is underutilized under the current allocation olicy [1]. Cognitive radio technology has been considered to mitigate this roblem [2]. It enables a secondary user to sense channel conditions and change its oerating characteristics to oortunistically access unoccuied rimary sectral bands. This new aradigm is tyically referred to as dynamic sectrum access (DSA). In the hierarchical model of dynamic sectrum access summarized in [3], users in the system are divided into a multitiered hierarchy where certain users have riority of channel access over others. Cognitive radio is conceived as a way for unlicensed secondary users (SU) to oortunistically access licensed sectral bands if rimary user (PU) activity is rotected from interference [4], [5]. This model is necessary because users will not agree to accommodate secondary networks to their own detriment. Therefore, a design imerative for a SU oortunistic access strategy is to minimize the SUs effect on PU transmissions. For examle, in the DARPA XG Manuscrit received July 23, 211. E. Jung is with the Deartment of Electrical and Comuter Engineering, University of California, Davis, 1 Shields Avenue, Davis, CA eajung@ucdavis.edu. X. Liu is with the Deartment of Comuter Science, University of California, Davis, 1 Shields Avenue, Davis, CA liu@cs.ucdavis.edu. Fig. 1. Tx(PU1) Rx(PU2) Rx(SU) Tx(SU) Rx(PU1) Tx(PU1) (a) SU interfering with two PUs that are satially nonoverlaing. Rx(PU2) Tx(PU1) Tx(SU) Channel 2 Channel 1 Rx(PU1) Channel 1,2 Tx(PU2) Rx(SU) (b) SU accessing 2 PU channels simultaneously. SU in environments with multile non-interfering PUs. roject [6], one of the three major test criteria in a cognitive radio rototye field test is to cause no harm [7]. Predictably, this is also one of the main bottlenecks of SU erformance. In this work, we consider acket collision robability as the PU rotection requirement. Under this requirement, the SU must guarantee that the acket collision robability of a PU acket is less than a certain threshold secified by the PU. This tye of constraint has already received some attention [8], [9], [1], [11]. In articular, we investigate SU erformance in a system with multile PUs with different acket collision robability constraints and usage atterns. There are common situations where multile PU systems may need to be rotected that have different owners, riorities, and usage atterns. Two simle examles are cases where the SU interferes with multile PUs which are satially or sectrally non-overlaing, as deicted in Fig. 1(a) and 1(b). Our aer makes the following contributions. First, we derive the otimal olicy for SU access in multile-pu systems with stationary general idle time distributions. We focus on a class of stationary access olicies, i.e. the same olicies are alied every time the channel becomes idle. We also look secifically at multile PU systems with exonential idle

2 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 2 time distributions, as well as idle time distributions that result in time-threshold otimal olicies. Finally, we extend our model to include two side information cases where extra information of PU activity is available to the SU. In the first case, the SU knows which PU is the last to become idle before an SU can transmit. In the second, the SU knows how long every PU has been idle before the channel becomes available. We investigate how such knowledge can affect the otimal olicy and erformance of the SU. The aer is organized as follows. In the next section, we resent some related work. We resent our system model in Section III and define the objective function in Section IV. In Section V we derive an otimal transmission scheme for a channel with multile PU constraints, where each PU has a general idle time distribution, and show that for certain idle time distributions, the otimal SU access olicy can be found in closed form. The system model is extended in Section VI such that the SU has side information on PU activity, and the otimal olicies with this new information are derived. In Section VII, we comare and analyze the erformance of our different olicies under the time caacity metric, and conclude our aer in Section VIII. II. RELATED WORKS In recent years, there has been an exlosion of research in cognitive radio. A large ortion of this research has been in sectrum overlay, where rotocols are devised to maximize SU sectrum utility when PUs are idle and rotect PU communication when they become busy. Within this aradigm, there are two focuses, satial and temoral domain research [3]. In the former, SU activity is assumed to occur in a much faster timescale than the PU activity, and hence the sectral environment (i.e. PU channel occuancy) is treated as static. The main roblem then becomes channel allocation between multile SUs given certain toologies, different channel availabilities, and interference between SUs. In [12], [13], [14], [15], the interference between SU nodes is modeled as a conflict grah, with varying methods and metrics used to allocate channels. In [16], [17], the authors formulate channel selection as a mixed integer linear rogramming roblem, under constraints on both ower and channel availability. In [18], the tradeoff between channel switching and maximizing SU bandwidth is considered under this assumtion. In all these works, PU activity is rotected since each SU node only has access to idle PU channels. Our work does not fall in this category since we assume that PUs change states in a timescale similar to that of SU activity. In the temoral domain category, PU activity varies quickly in the time domain and SUs within interference range must devise sensing and access schemes in concert to avoid significantly harming PU communication. As such, the metric used to measure interference to PUs is crucial. Several aers consider this ower, referred to as interference temerature, as the key metric. For examle, in [19], the authors consider multile SUs oerating in a multi-pu system where each PU has an average rate requirement and outage robability constraint, both functions of the interference ower caused by SUs. Power control for different states of PU activity is considered in [2]. However, in 27 the FCC officially ended consideration to establish an interference temerature standard for cognitive radio [21]. Their decision was based on several comments redicting a likely increase in interference with PUs in bands where it is used, stemming from the technical difficulties of imlementing such an aroach. Since we consider acket collision robability as our main constraint, we assume that any overla of SU and PU activity in the same band results in collision with the PU, which is a more conservative measure of PU interference. Like this work, other works have also considered acket collision robability. Several works have develoed medium access schemes for SUs under this rotection requirement [8], [9], [1]. One common formulation assumes a slotted system and is formulated using the Partially Observable Markov Decision Process (POMDP). For examle, [22], [23], consider a slotted structure network with a single PU rotection metric. Otimal access decisions are made by considering long observation history. Likewise, in [24], the authors consider a slotted system with a single CR, and model the roblem as a multi-armed bandit roblem to decide the best channel(s) to sense and access. In [11], the authors consider an overlay SU network on a multile PU network with slotted structure, where PU access deends on Markovian evolution. Our work is different from all of these because we consider multile different PU constraints that must be satisfied simultaneously. Many of these works also assume slotted activity, while our model is more general and can accomodate slotted and unslotted systems. Other works have considered unslotted systems as well. In [8], the authors introduce the erformance metric of time caacity, the average roortion of time that a SU can transmit without violating the PU s acket collision constraint, and generalize the results in [25]. The authors also extend their work to examine the imact of imerfect sensing on SU erformance as well as access issues between multile SUs in a single PU system. Our work in [26] generalizes the results for a single SU in [25] to the multile PU case for certain distributions, in articular for exonential distributions and distributions which result in time-threshold olicies. Our work in this aer resents a framework that can be used for general idle time distributions for multile PUs, and considers the effects of extra information about the PU system available to the SU. Much of the work in [25] can be generalized in this work as well. III. SYSTEM MODEL In this section we layout the model for PU and SU activity. Generally, we consider an SU that oerates within the interference range and on the same channels as multile noninterfering PU networks. Because the PUs are non-interfering, simultaneous transmission of multile PUs does not result in collision between them. A. Primary User Model In our system model, we assume there are M PUs that are indeendent and non-cooerative with the SUs and with

3 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 3 PU1 where PU2 V1 c i = lim Ni c, T N i PU V2 r and the PU rotection requirement is thus Sense Sense V =min(v1, V2 ) r c i η i, i 1,...,M}. (2) SU Collision w/ PU1 : Sense : Collision Collision w/ PU2 Collision w/ PU2 t We assume that the collision constraints for all PUs are known to the SU a riori. These constraints must be satisfied for all PUs. Fig. 2. Timing diagram for PU/SU system model. each other. A PU accesses its channel without sensing. PUs are non-interfering among themselves, i.e. they can transmit at the same time. This can be due to satial reuse as in Fig. 1(a), or due to communication on searate non-overlaing channels, as in 1(b). We assume acket-based transmission for all PUs in this aer. Fig. 2 illustrates the activity of each user. For PU i, V i denotes the idle time, which is governed by df/cdf f Vi ( )/F Vi ( ). Let E[V i ] = v i. The number of ackets transmitted er busy eriod is random and indeendent of PU i s idle times, and is denoted as N i, with E[N i ] = n i. We assume that the acket length is the same for all users, and normalize all activity to the length of a PU acket. Therefore, a busy eriod of the ith PU is N i. The robability that each PU is idle is then α i = vi v i+n i. All PU traffic is assumed to be stationary and ergodic. This is reasonable for acket-based data traffic, where the timescale for acket lengths is on the order of milliseconds but arrival rates are commonly stationary on the scale of hours [27]. We also define terms related to the union of the activity of all PUs in the system. We refer to this activity as unionized PU activity or simly the unionized PU. This activity rocess is also assumed to be stationary and ergodic. We denote the idle time of the unionized PU as V, which is governed by a robability distribution function f V ( ), with E[V ] = v. A busy eriod of the unionized PU is denotedn, with E[N ] = n Ṫhe robability that the unionized PU is idle is then α = v v +n = α i, (1) where the final equality results from the indeendence of the PU activity. The relationshi between each PU s activity and the unionized PU activity will be investigated in greater detail in Section IV. Each PU i also has a acket collision robability requirement denoted η i, defined as the maximum allowable robability of collision for a acket of the ith PU. Over a time interval [, T], we denote the number of ackets transmitted by the ith PU as N i, and the number of collisions exerienced by that user as Ni c. The collision robability of the ith PUs ackets exeriencing collision is denoted as c i = Pr[acket collision of ith PU], B. Secondary User Model Throughout this work we assume that there is a single SU, which may be a single CR radio or a SU basestation oerating within the range of multile PU networks [28]. We note that SUs may have access to multile bands, but low-cost SUs may have a set bandwidth requirement, and may not be able to frequently switch over channels at the timescales we are concerned with [25]. In [29], for examle, the authors state that channel switching times are often dwarfed by the minimum time of SU demand, and there is a risk of inefficiencies such as latency in switching channels too often. In that same work, a minimum allocatable bandwidth is a key factor in their model, which could lead to cases similar to the examle resented in Fig. 1(b) regardless of SU sensing and switching caabilities. Therefore, we assume that the SU is on a fixed channel and cannot transmit on a sub-band of the channel. We reiterate here that we are concerned with stationary otimal olicies. A stationary olicy is one that is alied whenever the channel becomes idle. We have: a) Packet Length: The SU slot length is denoted, and we assume v i, n i, and 1 (i.e. the length of a PU acket). In this aer, we study the extreme case where. In [8] it was roven that this results in the best SU caacity in the case of no overhead cost for a single PU channel, and overhead techniques used in that aer can be alied to our work as well. This assumtion simlifies analysis greatly, and our simulations show that non-zero acket lengths result in negligible differences. b) Sensing: We assume erfect sensing by the SU, i.e. that the SU can always detect the resence or absence of a PU, and that sensing time is negligible. The SU follows the listen-before-talk (LBT) rincile, where the SU senses the channel in each slot before allowing transmission. We assume that sensing occurs over the entire band of interest. The multile channel sensing roblem is a significant challenge and beyond the scoe of this work [28]. Our revious works consider imerfect sensing [8], [25], and these results aly here as well. We can also consider a fixed non-zero sensing time in the case of non-zero slot lengths. A fixed non-zero sensing time can reresent many commonly roosed sensing schemes, such as matched filter, energy, and feature detection. Since we assume a LBT scheme, the erformance in nonzero sensing cases is essentially a fixed fraction of the otimal erformance.

4 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 4 f Vi (t) F Vi (t) f r V (t) i FV r i (t) f V (t) F V (t) n i n α i α si f si(t) F si(t) h i(t) φ ij(t) TABLE I NOTATION FOR AN M-PU SYSTEM PDF of PU i idle time CDF of PU i idle time PDF of PU i residual idle time CDF of PU i residual idle time PDF of unionized PU idle time CDF of unionized PU idle time PU i average number of ackets er transmission average length of busy eriod for unionized PU Probability PU i idle Probability all PUs idle Probability PU i starts the idle eriod Idle time PDF given PU i starts the current idle eriod Idle time CDF given PU i starts the current idle eriod Portion of f V (t) due to PU i ending current idle eriod of length t Portion of f si(t) due to PU j ending current idle eriod of length t c) Collision Detection: Whenever an SU and PU transmit simultaneously, we assume that both exerience collision and the SU can detect the collision after the transmission. Perfect sensing, the SU acket length, and the LBT assumtion ensure that acket collisions occur only if a PU accesses the channel while an SU is already transmitting. This ensures that for any single PU busy eriod, at most one PU will exerience acket collision. Assuming erfect sensing and collision detection allows us to focus on investigating otimal caacity. Collisions with PUs are demonstrated in Fig. 2. d) Knowledge of Individual PUs: We assume that the SU has knowledge of the collision constraint, the idle time distribution, and the mean busy time of each individual PU a riori, i.e. f Vi ( ), n i, η i for all PUs, which are indexed by i. This knowledge can be obtained from the network usage histories obtained from network oerators [27]. This is feasible deending on the conditions of SU deloyment, where SU and PU network oerators may cooerate during initial SU network lanning and deloyment. For examle, in the DARPA rogram both PUs and SUs are military, so lanning of SU networks could conceivably include knowledge of PU network statistics [25]. e) Performance Metric: The SU s erformance metric is the time caacity, the ercentage of time that the SU can transmit successfully under the collision robability constraint. This metric is defined as below: C s = lim T SU s successful access time in [,T]. (3) T Since the channel observed by the SU has idle robability α, clearly C s α. We show through simulations later that a system with non-zero slot-length and sensing time results in a fraction of the otimal C s corresonding to the fraction of the slot time where transmission occurs. IV. OBJECTIVE FUNCTION The SU objective is to maximize its time caacity while satisfying all PU s collision constraints, max q Q subject to C s (q) c i (q) η i, i 1,...,M, (4) PU1 PU2 PU Fig. 3. V1 V2 r V =min(v1, V2) r r V1 V2 r V =min(v1, V2) Timing diagram showing two tyes of idle eriods. whereqis the set of all ossible transmission olicies that can be selected by the SU, while C s (q) and c i (q) are resectively the time caacity and robability of collision with user i given a olicy q( ) Q. Based on the system model, we now define the form of all olicies q( ) and derive the resentation of the objective function. The derivation assumes the system consists of a single SU co-existing with M non-interfering PUs. All notation is summarized in Table I. The objective function requires the derivation of the idle time distribution for the unionized PU activity. For brevity, we defer this derivation to Aendix A, and define the necessary terms here. Let t be the time elased since the beginning of the most recent idle eriod. We denotef si (t) as the idle time PDF of the unionized PU given that PUiis the last to sto transmission. In this case we say that PU i starts the idle eriod that follows. Then the unionized PU idle time distribution f V (t) can be rewritten as f V (t) = si f si (t), (5) where si is the robability that PU i starts an idle eriod. From Fig. 3, we can see that the residual idle time Vj r of PU j is also imortant to the system. The residual idle time distribution f V r j (t) is f V r j (t) = 1 F V j (t) = 1 F V j (t). (6) E[V j ] v j From the same figure, we observe that for M users, if PU i starts the idle eriod, V = min(v r 1,V r 2,...,V r i 1,V i,v r i+1,...,v r M ). We define φ ij (t) as φ ij (t) = f V r j (t)[1 F Vi (t)] [1 F V r k (t)], (7) k=1 k i,j which can be described as the ortion of the conditional PDF f si (t) due to PU j transmitting first after an idle eriod of length t. We note that f si (t) can now be defined as f si (t) = φ ij (t). (8) t

5 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 5 We also define h i (t) as h i (t) = sj φ ji (t), (9) which can be thought of as the ortion of the idle time PDF f V (t) that accounts for PU i transmitting first after an idle eriod of length t. We can rewrite f V (t) as f V (t) = h i (t). (1) We note here that, under the assumtion of ergodicity of the unionized PU activity, the equations for f V (t) can only be said to hold almost everywhere for an M PU system. Therefore, f V (t) and the resulting otimal olicy can only be said to aly with robability one instead of for all outcomes given a set of PU distributions. In all of the simulations in Section VII, all simulated instantiations of unionized PU activity and the resulting otimal olicies hold to the derivations made here, and we omit this fact for the sake of brevity in all following discussions. We now derive the objective function for our system. We define Φ (t) as the channel state of the unionized PU: Φ Idle, if all M PUs idle (t) = Busy, otherwise. We also define the general form of the SU olicy q(t) as the robability that the SU olicy transmits at time t, (t), if Φ (t) = Idle q(t) =, otherwise, where (t) 1. We note that this is a stationary olicy, where the same olicy is alied in every idle/busy cycle. Within each idle/busy cycle, the olicy is deendent on t, which reresents the time since the beginning of the most recent idle eriod. The structure of the otimal olicy q ( ) is the main focus of our work. For an SU olicy q( ), the time caacity equation (3) can now be defined as C s (q) = t f V (t) q(τ)dτdt v = G s(q) +n v, (11) +n where G s (q) = t f V (t) q(τ)dτdt. The numerator in Equation (11) calculates the average time transmitted in an idle/busy cycle of the unionized PU, while the denominator is the length of the average idle/busy cycle. The SU can collide with PU i only when that PU is the first to transmit following an idle eriod. Therefore, defining Ψ i (q) = q(t)h i (t)dt, from Equation (9) the constraint η i for PU i is satisfied under olicy q(t) if Ψ i (q) n i η i v +n v i +n i. (12) This equation can be understood as follows. Ψ i (q) is the robability that an SU using olicy q( ) will collide with a single acket from PU i in any given idle/busy cycle of the unionized PU with average duration v +n, as h i(t) is the ortion of the unionized PU idle time distribution function for when PU i is the first to begin transmitting after an idle eriod. At the same time, n i η i is the robability of collision with the SU that PU i can tolerate under its collision constraint for any given idle/busy cycle of PU i with average duration v i +n i. Therefore, a olicy q( ) must be determined that satisfies n i η i weighted by the ratio of these two different idle/busy cycle lengths. We now define η i = n v iη +n i v i+n i, and restate the objective function defined in (4) for the SU as max q(t): q(t) 1 C s (q) subject to Ψ i (q) η i, i 1,...,M}. (13) We note that the objective function (13) is in fact a convex otimization roblem. This is because the function C s (q) and the constraints Ψ i (q) from (13) are integral functions of q( ), where q(t) [,1] t, and integration is a linear function which thereby reserves convexity [3]. V. OPTIMAL POLICY FOR MULTIPLE PUS WITH GENERAL IDLE TIME DISTRIBUTIONS We now derive the otimal stationary SU access olicy in an M-PU environment with general idle time distributions. We first derive the most general form of the otimal olicy q ( ). We then resent a search algorithm for determining the otimal olicy based on rinciles of convex otimization. A. Deriving the General Otimal Policy We consider the following olicy q( ), defined as 1 F V (t) 1, if > M µ 1,Φ (t) = Idle ih i(t) 1 F q(t,µ) = V (t), if M = µihi(t) 1,Φ (t) = Idle, otherwise, (14) where µ = [µ 1,µ 2,...,µ M ] T. The condition for the iecewise boundaries of q(t, µ) resemble an inverted hazard function, where the df f V (t) is relaced by the sum of its h i (t) comonents weighted by µ. Each µ i term can be thought of as the imortance given to each PU s collision constraint. We then define the otimal olicy q (t) as q (t) = q(t,µ ), (15) where µ = [µ 1,µ 2,...,µ M ]T. Both µ and are chosen such that the following conditions are satisfied for i 1,...,M}: µ i, (C1) µ i = if q (t)h i (t)dt η i, q (t)h i (t)dt < η i. (C2) (C3)

6 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 6 Note here that µ i can be exlained as the Lagrange multilier for the ith constraint, andq ( ) is the function to be otimized. We now state the following theorem. Theorem 1: Policy q ( ) is an otimal olicy that maximizes the SU throughut while satisfying the collision constraints of all PUs. Proof: Consider any feasible olicy q( ) that satisfies the collision robability constraints of all PUs. We now rove that G s ( q) G s (q ), and therefore C s ( q) C s (q ). From (11), G s ( q) = f V (t) (a) G s ( q) = q(t) + µ i η i (b) + [ q (t) µ iη i = G s (q ) = G s (q ). t µ i q(τ)dτdt ( ) q(t)h i (t)dt η i 1 F V (t) [ 1 F V (t) µ i ] µ i h i(t) dt ] µ i h i(t) dt ( ) q (t)h i (t)dt η i The inequality (a) is true because q( ) must be a feasible olicy, which results in a collision robability less than or equal to the constraint. The inequality (b) results becauseq ( ) follows conditions (C1), (C2), and (C3), and q ( ) is ositive whenever 1 F V (t) M µ i h i(t) is ositive. We note that this olicy derivation is similar to that of the Lagrangian method [3]. However, it is by itself a roof, and does not make use of Lagrangian duality. Therefore, the otimal olicy search is in fact a search for vector µ such that the otimality conditions (C1), (C2), (C3) are satisfied. Requirement (C1) can be imosed by searching over only non-negative values for µ, but the others are deendent on q(t, µ). B. Otimal Policy for M PUs with Exonential Idle Times It is a common assumtion that channel idle time is distributed exonentially. For examle, in [27], the authors erform a measurement study that shows that the exonential call arrival model is adequate for cellular networks. It was also shown in [25] that the exonential case also rovides a lower bound to the achievable time caacity of any PU system. The same result, as we will show, alies here also. Therefore, in a system where the PUs average idle times are known but the idle time distributions are not, the olicy derived here can be used while guaranteeing the acket collision robability constraints of all users. When all M PUs have exonential idle time distributions, we can show that the otimal olicy q ( ) is q, ifφ (t) = Idle (t) = (16), otherwise, where [,1] is determined by the constraint conditions. In such a system the SU has equal robability of transmitting any time the channel is idle, so the time caacity isc s = α. This comes from the idle time distributions. Since all distributions are memoryless, f V r i (t) = f Vi (t), and F Vi (t) = F V r i (t) = 1 e λit for all i. Since V = min(v1 r,v 2 r,...,v i,...,vm r ), This also means that F V (t) = 1 e ( M λi)t. (17) h i (t) = C i e ( M λi)t, i 1,...,M}, (18) where C i is some constant. From (17) and (18), we observe that for any µ, there are three ossibilities: 1. 1 F V (t) < M µ ih i (t) for t F V (t) > M µ ih i (t) for t F V (t) = M µ ih i (t) for t. If the first is true, then from (15) we have q(t) = for all t, while if the second is true, q(t) = 1 for all t, imlying that the collision constraints are trivial. Therefore, assuming nontrivial collision constraints, an aroriate µ satisfies condition three, and (15) reduces to (16). Therefore, determining µ is not necessary, because we only need. We now determine a that satisfies (2) for all PUs. We note that can be found by alying (16) and (12), so we resent an intuitive exlanation here. Observing that is the robability that a single collision occurs during a single cycle of the unionized PU activity, this means a collision with PU i only occurs if the other M 1 PUs are idle when PU i begins transmitting. Therefore: j i α j = α α i n i η i, where the right side results from the fact that only one acket collision can occur when any PU becomes busy again, and on average PU i transmits n i ackets er busy eriod. It follows that can be written: = min (α i i 1...M} α n iη i,1). (19) Given this result for, we obtain the following theorem. Theorem 2. The maximum time caacity of the SU in a channel with the unionized activity of M PUs with exonential idle time distributions is: C s = min i 1...M} (C si,α ). (2)

7 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 7 PU1 PU2 V1 where Ti is determined by T i h i (τ)dτ = η i. (23) PU SU Fig. 4. PU1 Protects PU2 PU2 Protects PU1 Collision Protection with 2 PUs. t V2 r V =min(v1,v2 ) r PU2 Protects PU1 where C si = α i n i η i, which is the time caacity that can be achieved in the single-pu system consisting of the ith PU with exonential idle time. Proof: This results directly from (19) and (11). Another roof of this theorem is also available in our earlier work [26]. This result has the following intuitive exlanation. As Fig. 4 demonstrates, the PUs rotect each other from collision. This reduces the collision robability of each PU, allowing the SU to transmit more aggressively. This offsets the reduced channel availability. In some cases, all PUs will be rotected to the oint that their collision robability constraints cannot be violated, and the SU will transmit whenever the channel is idle, resulting in = 1 with time caacity C s (q ) = α. In addition, we also note that can be calculated simly using (19), which greatly reduces calculation time. This mutual rotection also benefits PU systems with general idle time distributions, but does not result in the same simlified structure as the exonential case. C. Time-Threshold Policy There are many cases in which the PU idle distributions result in a time-threshold olicy. For examle, systems where a PU has a uniform idle distribution or Weibull distributions with shae arameter under unity commonly result in a timethreshold olicy. Therefore, we investigate this case here. A time-threshold olicy begins transmitting as soon as the channel becomes idle for a length of time T, or until the channel becomes busy: q 1, if t T,Φ (t) = Idle (t) = (21), otherwise. We now state the following theorem. Theorem 3. If 1 F V (t)/h i (t) can be shown to be monotonically decreasing for i = 1,...,M, then the otimal olicy is a time-threshold olicy of the form in (21). Furthermore, T is T = min(t1,t 2,...,T M ), (22) t Proof: The condition for this olicy results directly from (15), which can be rewritten as 1 F V (t) 1, if > M q µ i hi(t) 1,Φ (t) = Idle (t) = 1 F V (t), if M µ = i hi(t) 1,Φ (t) = Idle, otherwise. Clearly, if the stated condition is satisfied, the above equation reduces to (21). Ti can be interreted as the otimal timethreshold for the SU if only PU i is constrained. Since only the minimum Ti can guarantee that all PU constraints are satisfied, the otimal olicy follows (21) with T as defined by (22) and (23). This leads to significant comutation reduction in determining the otimal SU olicy, since the olicy can now be found through (23). As stated before, several common distributions result in this tye of olicy. D. Otimal Policy Search Because the olicy search is essentially an otimization over the Lagrangian multilier vector µ, we can use convex otimization techniques to search for the otimal olicy. Therefore, in our numerical results, we use a descent method with line search to determine µ [3]. The idea of the olicy search is that a search or descent direction is chosen based on a current choice of µ, a distance to search in that descent direction is chosen based on a line search algorithm, and µ is udated. This rocess reiterates until a stoing criterion is reached. For ease of discussion, we define Ψ i (q) and L(q,µ) as Ψ i (µ) = L(µ) = G s (q(t,µ))+ q(t,µ)h i (t)dt, µ i (η i Ψ i(µ)). Ψ i (µ) is the collision robability achieved with PU i given olicy q(t, µ) from (12), and L(µ) is essentially the Lagrangian dual for the objective function (13). Beginning with an arbitrary, feasible choice for µ, the algorithm first determines a search direction µ using the gradient descent method, µ = [Ψ 1 (q(t,µ)) η 1,...,Ψ M (q(t,µ)) η M]. This direction is in fact the negative gradient of L(q,µ) with resect to µ. This direction increases the µ i corresonding to violated constraints under the current olicy, and decreases the µ i for PU constraints which are obeyed. This can be analogized as giving more/less weight to the stricter/looser PU constraints in the search. Next, a line search algorithm is used to determine the ste size ν, the distance to increment in the search direction. In

8 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 8 our numerical results we use a modified backtracking line algorithm [3]. It is essentially the same as the standard algorithm, but with conditions that revent the resulting ν from searching into negative µ i values which would violate condition (C1). Finally, µ is udated as µ µ+ν µ. If the udated µ satisifes conditions (C2) and (C3) to within a reasonable limit, the algorithm is terminated with the current iterate as µ. In ractice, this algorithm runs quickly. For all results in Section VII, each 2-PU-system olicy is found in under.3 seconds (and in many cases, significantly less) running on a Pentium GHz rocessor with 1 GB of RAM. Given that any olicy is meant to be oerative on the order of hours, this time is negligible. VI. PU SIDE INFORMATION We now investigate how extra information of PU activity can affect the otimal olicy, and ultimately the time caacity of the SU. In addition to the original system model assumtions, this information is assumed to be available to the SU through augmented sensing caabilities from the original system model. These two cases of extra PU information are as follows: Side Information 1: The SU knows which PU was the last to transmit. Side Information 2: The SU knows which PU was the last to transmit and how long the other PUs have been idle rior to the unionized channel going idle. We use SI-1 and SI-2 to describe these two cases, and No-SI to refer to the original olicy laid out in Section V. We note that in general the sensing caability of SI-1 would seem to imly the caability of SI-2, i.e. if a sensor can differentiate PUs, it is easy to assume it can also kee track of how long each has been idle. SI-1 and SI-2 can achieve significantly better erformance than the original case. Through our derivations we show that SI-2 is comutationally intractable. We now derive the new otimal olicy for each case. A. Otimal Policy for SI-1 We wish to leverage the new information when determining the otimal olicy q (t) of the SU. Naturally, the SU can act differently based on which PU ends transmission last. Therefore, an SU olicy q(t) can be written as q(t) = q i (t), if PU i is last to transmit, Φ (t) = Idle, otherwise. (24) where q i (t) is the SU access olicy when PU i is the last to transmit. From (12) and (13), the time caacity and the collision constraint can be rewritten as M C s (q) = si q i (t)[1 F si (t)]dt v +n, (25) sj q j (t)φ ji (t)dt min(η i,1). (26) where F si (t) is the cdf of the idle time distribution given that PU i is the last to transmit. Using these more secific equations, the objective function still follows the form of (13). We can see that the otimal olicy q (t) is similar to (14), excet that there are now M olicies for when each PU is the last to transmit. Therefore, we now define the otimal olicy qi (t,µ) given that PU i is the last to transmit for a given multilier vector µ: qi (t,µ) = 1, if 1 F si (t) > M µ jφ ij (t),φ (t) = Idle, if 1 F si (t) = M µ jφ ij (t),φ (t) = Idle, otherwise. (27) We note that this equation is quite similar to (15). F V (t) is relaced with F si (t), and h i (t) is relaced with φ ij (t), since the olicy being defined is only oerative when PU i is the last to transmit. We now define the otimal olicy q (t) as q (t) = q i (t,µ ), if PU i is last to transmit,φ (t) = Idle, otherwise, (28) where µ satisfies the conditions laid out in Section V-A. The roof of the otimality of the olicy is analogous to that of Theorem 1. This olicy can be obtained using a slightly modified version of the algorithm resented in Section V-D. B. Otimal Policy for SI-2 We now derive the otimal olicy for the SI-2 case. As stated reviously, this case is comutationally intractable and rimarily serves as an uer bound for ossible erformance for stationary olicies under our model. In the SI-2 case, the SU knows how long every PU has been idle rior to the channel going idle. We call this time the lost time, denoted by X i for the ith PU, with x i as a realization of X i. By definition, at least one PU s lost time will be zero at the start of any idle eriod, since at least one PU is transmitting rior to the moment the unionized PU goes idle. We define the vector X i as the vector containing all of the lost times given PU i is the last to return: X i = X 1,...,X i 1,,X i+1,...,x M } and x i = x 1,...,x i 1,,x i+1...x M } as a articular instantiation of X i. We note that the ith element is always zero for x i. The SU olicy deends on both the PUs idle time distributions and the vector X i. We denote the SU access olicy given that PU i is the last to transmit with lost time vector X i as q(t X i ), and a SU olicy q(t) as q(t) = q(t x i ), if X i =, X i = x i,φ (t) = Idle. (29) The time caacity and constraint equations (11) and (12) must be re-derived using the idle time distribution conditioned on the values of the lost time vector X i.

9 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 9 To obtain the time caacity and constraint equations, we must determine several related dfs conditioned on the lost time vector. We first define f si (t x i ), the conditional df of the idle time given that PU i is the last PU to transmit with lost time vector x i, f si (t x i ) = φ ij (t x i ). (3) In this case, φ ij (t x i ) is the ortion of f si (t x i ) due to PU j ending an idle eriod of length t, given that PU i is the last to transmit with a lost time vector of x i. We refer the reader to Aendix A for the derivation φ ij (t). If j = i we have If j i, we have: φ ii (t x i ) = f Vi (t) φ ij (t x i ) = f V j (t+x j ) 1 F Vj (x j ) j i k=1 k j 1 F Vj (t+x j ). 1 F Vj (x j ) (31) 1 F Vk (t+x k ). 1 F Vk (x k ) (32) We must also determine the distribution of the lost time vector X i. The distribution for X i is the same as the residual time distribution for the ith user: f Xi (x i ) = f V r i (x i ) = 1 F V i (x i ) v i, (33) and the distribution of X i is the joint distribution of all of the lost time distributions of the other variables. Since all PUs act indeendently, this distribution amounts to f Xi (x i ) = f Xj (x j ), (34) j i wheref Xi (x i ) is not included in (34) because inx i,pr[x i = ] = 1. With (3), (31), (32), and (34) we can derive the equation for f si (t) as f si (t) = f si (t x i )f Xi (x i )dx i. (35) x i The equation for f V (t) still follows (5). Integrating (3) over t also allows us to obtain the conditional cdf F si (t x i ). We also rewrite h i (t) in terms of φ ij (t x i ) as h i (t) = sj φ ji (t x i )f Xi (x i )dx i. (36) x i We are now able to derive the time caacity C s (q) and constraint equations for SI-2: q(τ)[1 F V C s (q) = (τ)]dτ v +n M = si x i q(τ x i )[1 F si (τ x i )]dx i dτ v +n, (37) sj x j q(t x j )φ ji (t x j )f Xj (x j )dx j dt η i. (38) The objective function then follows the form of (13) using the time caacity and constraint equations (37) and 38). As in the revious cases, we define the otimal olicy q (t) as the olicy that corresonds to the vector µ such that the otimality conditions (C1)-(C3) are satisfied. The otimal olicy q (t x i ) for PU i given x i is: q (t x i ) = 1, if 1 F si (t x i ) > M µ j φ ij(t x i ),Φ (t) = Idle, if 1 F si (t x i ) = M µ j φ ij(t x i ),Φ (t) = Idle, otherwise, (39) and the otimal olicy q (t) is q q (t x i ), if X i = and X i = x i, Φ (t) = Idle (t) =, otherwise, (4) where µ = µ 1,...,µ M } satisfies (C1)-(C3). The roof is the same as that resented in theorem 1. This olicy gives better erformance than either the No-SI or SI-1 cases because of the additional information. However, in ractice, obtaining the otimal olicy in (4) is comutationally difficult. For any µ, the collision robability deends on an infinite number of olicies q(t x i ) corresonding to all ossible lost time vectors, rendering numerical solutions rohibitively exensive. VII. NUMERICAL RESULTS We now resent numerical results demonstrating how SU erformance is affected by the number of PUs in the system and the side sensing information defined in the revious section. The results resented are meant to demonstrate the effect of time caacity in a number of different network scenarios where multile PUs may need to be rotected. Therefore, we leave out secific details of PHY/MAC layer network scenario assumtions. First, we investigate how the number of PUs affects the SU erformance for the No-SI case. We show that while time caacity generally degrades with the number of users, throughut increases to a oint if the PUs are on different channels. Then we show that in the SI-1 and SI-2 cases the extra PU information available to the SU over the No-SI system model leads to higher time caacity while still satisfying PU acket collision robability constraints. A. Multile PUs We first investigate how multile PUs affect SU erformance in the No-SI system model. In all simulations in this section, we have multile PUs with the same idle/busy time distributions,v i = 8 andn i = 2, and the same collision constraint η. We comare SU erformance for two different idle time distributions, exonential and uniform. The PU acket length is 1, and the SU acket length =.1. Therefore, any SU olicy q( ) must assign transmission robabilities for time

10 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 1 C s C s vs. Multile Channels, v i = 8, n i = 2, α =.8 Uniform Idle, η=.1 Exonential Idle, η=.1 Uniform Idle, η=.2 Exonential Idle, η=.2 Uniform Idle, η=.3 Exonential Idle, η=.3 α Fig. 6. Throughut vs. Number of PU Channels (M) with v i = 8,n i = 2. R s vs. M PU Channels, v i =8, n i =2, η=.1 R s P/N =1, unif P/N =1, ex P/N =1, unif P/N =1, ex Number of Channels Fig. 5. C s vs. Multile PUs, with single PU arameters v i = 8,n i = 2. increments of length, and all idle time distribution functions are calculated with time increment of also. First, in Fig. 5, we study the time caacity erformance as a function of the number of PUs to be rotected. This is an accurate measure of SU erformance in the satial searation case demonstrated in Fig. 1(a), since all PUs and the SU oerate on the same channel. In Fig. 5, each curve reresents a simulation where all PUs have the same rotection requirement and idle time distribution, either exonential or uniform. We first observe that for all cases, the time caacity is eventually limited by the idle robability α. This corresonds to when the PUs are sufficiently rotected such that no PU constraint can be violated. In this case, the SU transmits whenever the channel is idle. We also note that the time caacity for uniform idle distribution cases degrades as the number of PUs grows, with the largest dro between M = 1 and M = 2. However, the exonential idle time cases exerience no degradation until the SU transmits with robability 1, corresonding to a time caacity of α. Finally, we observe that the uniform case outerforms the exonential case for all η and M before the time caacity is limited by α. This is because the memorylessness of the exonential case reduces the SU access olicy to a random access scheme with robability corresonding to the collision constraint. On the other hand, in the uniform distribution case, the time that the unionized channel has been idle hels in redicting when the channel will become busy. This results in a olicy that exloits this redictability, transmitting more aggressively earlier on in an idle eriod. In a case where each PU resides on a different channel as in Fig. 1(b), we must also consider the bandwidth of the unionized channel to evaluate SU erformance. We consider this in Fig. 6. In these simulations, we now assume that in addition to each PU having the same usage statistics and collision constraints as in Fig. 5, each PU occuies its own channel of bandwidth B. All channels have the same noise ower densityn, and the SU oerates only under a maximum ower constraint P. We now define the throughut of the SU R s (M) as the roduct of the time caacity and the Shannon R s R s M (a) η =.1 R s vs. M PU Channels, v i =8, n i =2, η=.2 P/N =1, unif P/N =1, ex P/N =1, unif P/N =1, ex M (b) η =.2 R s vs. M PU channels, v i =8, n i =2, P/N =1 η=.1, unif η=.1, ex η=.2, unif η=.2, ex η=.3, unif η=.3, ex M (c) η varying, P/N = 1 caacity of an SU oerating on M channels. Therefore, ) P R s (M) = C s (M)MB log 2 (1+, (41) MB N where C s (M) is the time caacity of the SU on M channels. Therefore, for any simulation, C s (M) always follows one of the curves in 5. For simlicity, we also assume that B = 1. In Fig. 6(a) and 6(b), η is held constant at.1 and.2 resectively, and each curve reresents a different P/N ratio and idle time distribution (exonential vs. uniform). First, observing Fig. 5, we see that when η =.1 and.2, the number of PUs that results in a time caacity C s α is 8 and 5, resectively. Now, from 6(a) and 6(b), we see that when η =.1 and.2, the maximum throughut also occurs at 8

11 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 11 and 5 PU channels resectively, regardless of distribution and SU transmission ower. This suggests that a good heuristic to obtain maximum throughut in a multi-pu multi-channel system is to use as many channels as ossible, such that the time caacity is not severely limited by the idle robability of the channel. Fig. 6(a) and 6(b) also show that the uniform case has a higher throughut than its corresonding exonential case until the maximum throughut is reached. This is again due to the memorylessness of the exonential case, which makes the arrival of the PU unredictable. However, we also notice that the exonential and uniform cases with the same SU ower and η value both result in roughly the same erformance at the elbow of each of the curves. The cause of this is obvious: in either case, the SU is transmitting whenever the channel is idle, and both cases have the same idle robabilities. This result suggests that a strong heuristic to obtaining a channel grouing that obtains near otimal erformance may be to treat all channels as exonential regardless of their actual distributions. This will significantly reduce comutation of the unionized PU idle time distribution as well as the otimal access olicy. Finally, in Fig. 6(c), P/N = 1 for all simulations, and each curve reresents a different η value and idle time distribution. We note that regardless of the η value, as the number of PU channels goes u, eventually erformance is limited by the time caacity. B. Comarison of No-SI and SI-1 We now comare the erformance of the No-SI and SI- 1 cases. In these simulations, activity of a two PU system is generated. Both PUs have uniform idle time distributions with v 1 = v 2 = 8, and general busy time distributions with n 1 = 2, n 2 = 4 ackets er transmission. Several busy time distributions were considered, including uniform, exonential, and deterministic acket length distributions, which all resulted in very similar results. We assume that the ackets for both PUs have the same length, so that α 1 =.8, α 2 =.6667, and α = PU and SU acket lengths are 1 and.1, resectively. Results with larger SU acket lengths, including a case where SU and PU acket lengths are equal, showed negligible differences in erformance with the cases resented here, and were therefore omitted. In Fig. 7(a), and SI-1 olicies are lotted as a function of η, where both PUs are assumed to have the same acket collision robability constraint, i.e. η 1 = η 2 = η. Table II(a) dislays the collision robability values obtained by the No-SI (NS) and SI-1 (S1) olicies for each PU and desired η value. In Tables II(a), II(b), and III, the constraint values set by the PUs are labeled as η i, and the collision robabilities achieved by the otimal olicies are labeled as ˆη i. We observe that the SI-1 otimal olicy achieves higher C s when η <.35. We also notice that, while both olicies are able to achieve the desired η 1 value, the SI-1 olicy is able to achieve higher collision rates than No-SI for PU2 without violating the collision constraints. This is because SI-1 exloits the extra information to transmit more aggressively, achieving a higher time caacity. Finally, at η =.35, we notice that both olicies achieve aroximately the same C s, and that both olicies have collision robability rates less than the desired η. This corresonds to the case where the SU transmits whenever the channel is idle: C s = α =.5333, and neither collision constraint can be violated. In Fig. 7(b), we observe the effect that differing collision constraint values have on the time caacity for No-SI and SI-1. Time caacity C s is lotted against changing η 2, while PU1 s collision constraint is held at η 1 =.1, with corresonding achieved collision robability values dislayed in Table II(b). Again it is clear that the SI-1 olicy erforms better in general, and similarly, SI-1 achieves a higher collision robability than No-SI without violating the collision constraints. The one excetion occurs at η 2 =.6, where both cases satisfy the collision constraints with equality (bolded in Table II(b)). The corresonding C s values achieved by No-SI and SI-1 are the same at this oint. In general, this observation holds true for all combinations of idle time distributions that result in T olicies: for any η 1 value, there is a corresonding η 2 value such that both constraints are satisfied with equality by the No-SI and SI-1 otimal olicies. In this case, the No- SI and SI-1 olicies are both time-threshold olicies with the same T value, which results in their corresondingc s values being equal. We also observe that both C s curves become constant at higher η 2 values. This is because η 1 becomes the limiting constraint for both olicies, such that neither No-SI nor SI-1 can transmit more aggressively to exloit looserη 2 constraints. Finally, in Table III, we observe the effect of non-zero SU slot lengths and sensing times on erformance. We run all olicies over the same simulated PU transmission atterns as in the results from Fig. 7, with v 1 = v 2 = 8, n 1 = 2, and n 2 = 4. PU slot length is again 1. We assume that the SU senses the channel for a constant amount of time in every slot, with s denoting the sensing time. The table in this case corresonds only to SI-1 olicies; the results from the No-SI case show similar trends. From the table, we see that erformance is affected very little by the increased acket length. In all cases with zero sensing time, when comared with the theoretical olicy erformance, there are only slight variations of achieved collision robabilities and time caacity. Generally, higher values of ˆη i corresond to slightly higher values of time caacity. These differences are minor, since the differences in ˆη i are less than 2% in all cases with resect to the =.1 theoretical case. In the cases with non-zero sensing time, we see that while achieved collision robabilities remain the same, C s decreases due to the time sent sensing the channel. The decreases in C s corresond to the s / factor sent for sensing in each time slot. Therefore, we conclude that for non-zero SU slot lengths, erformance remains within accetable limits assuming that the SU slot length is not greater than the PU acket length. C. SI-2 Policy Performance We now comare SI-2 erformance to the other two cases. Simulation arameters are the same as the revious section.

12 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 12 Fig. 7. Time caacity erformance for 2 PUs with uniform idle time distributions, v 1 = v 2 = 8, n 1 = 2,n 2 = 4. Time Caacity vs. η, η 1 =η 2 =η Time Caacity (C s ) vs. η 2, η 1 = C s.3.2 No SI SI 1 C s.15.1 No SI SI η 1 (a) Time Caacity vs. η = η 1 = η η 2 (b) Time Caacity vs. η 2 with η 1 =.1 TABLE II ACHIEVEDη 1,η 2,C s FOR FIG. 7 (a) Achieved values for Fig. 7(a), η = η 1 = η 2 η ˆη 1 (NS) ˆη 2 (NS) ˆη 1 (S1) ˆη 2 (S1) (b) Achieved values for Fig. 7(b), η 2 varying, η 1 =.1 η 2 ˆη 1 (NS) ηˆ 2 (NS) ˆη 1 (S1) ˆη 2 (S1) TABLE III SU PERFORMANCE FOR SI-1 POLICY WITH NONZERO SLOT LENGTH, SENSING TIME s η 1 =.1,η 2 =.6 η 1 =.1,η 2 =.1 Policy Tye C s ηˆ 1 ˆη 2 C s ηˆ 1 ηˆ 2 =.1, s = (Theoretical) =.1, s = = 1, s = = 1, s = = 1, s = For SI-2 simulations, otimal search is imractical because of heavy comutation. Instead, we use a randomly selected (µ 1,µ 2 ) air to generate an arbitrary, simlified SI-2 olicy, i.e. no olicy search is initiated. Since the SI-2 olicy is deendent on the lost time vector X i, for any (µ 1,µ 2 ) air there are an infinite number of olicies corresonding to different lost times, meaning that the achieved collision robability of any SI-2 olicy is comutationally exensive. To make this more tractable, we divide the ossible lost time into intervals of width W. For each interval, a olicy is determined in a manner similar to (39). For a 2-PU system with a given (µ 1,µ 2 ) air, q(t x i, x j W = k) = 1 F 1, if si(t x j=kw) > 1, M µ Φ (t) = Idle jφ ij(t x j=kw) 1 F, if si(t x j=kw) M µ jφ ij(t x j=kw) = 1, Φ (t) = Idle, otherwise, (42) where x i =, and x j is the lost time of PU j with j i. The modified olicy is then q(t x i, xj q(t) = W = k), if X i = x i, Φ (t) = Idle, otherwise. (43) Although the olicy defined in (43) reduces comutation greatly, it is still exensive to search for an otimal(µ 1,µ 2 ) for a given η 1,η 2 air. Instead, we obtain an arbitrary SI-2 olicy using a randomly selected (µ 1,µ 2 ) air. We then comare the erformance of this olicy to otimal No-SI and SI-1 olicies. Using the collision robabilities obtained under the SI-2 olicy, we erform the otimal olicy search to determine the No-SI and SI-1 otimal olicies corresonding to those collision robabilities. Table IV shows the results of one such comarison. We use a lost time interval width W = 1, and obtain a SI-2 olicy that yields η 1 =.11,η 2 =.11. The otimal search algorithm is run for No-SI and SI-1 using those constraints. The SI-2 olicy achieves a 9.2% gain over SI-1, and a 34.3%

13 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 13 TABLE IV PERFORMANCE COMPARISON BETWEEN 3 CASES, WITHη 1 =.11, η 2 =.11, AND LOST INTERVAL WIDTHW = 1. gain over the No-SI. Policy η 1 achieved η 2 achieved C s No-SI SI SI VIII. CONCLUSION In this aer, we studied oortunistic sectrum access in a system with multile PUs, where the PUs are heterogeneous in terms of idle time distribution and acket collision robability requirement. We determined the form of the otimal olicy, and secified two secial cases for which the otimal olicy is easier to determine: the case where all PUs have exonential idle times, and the case where the PU idle time distributions lead to a time-threshold olicy. We then studied the effect that extra information of PU activity has on the SU otimal olicy. The first case studied was one in which the SU knows which PU was the last to transmit before the channel becomes idle. In the second, the SU had the additional knowledge of how long each PU had been idle before all PUs became idle. For both cases, otimal olicies were also determined. Simulations results demonstrated that the extra PU information significantly increases SU erformance. These results have imlications for future work in channel ooling and allocation in SU networks. One such roblem is how an SU should otimally access a set of PU channels because there is a tradeoff between instantaneous data rate and time caacity. Another roblem is channel allocation for multile SUs. In general, revious work on allocation has assumed a static sectral environment, i.e. licensed channels are idle or busy for long eriods of time. This work imlies new allocation scenarios in which PUs have frequent idle/busy transitions, and SUs must consider PU activity statistics when allocating channels since certain combinations of channels may be better groued together. APPENDIX A DERIVATION OF UNIONIZED PU IDLE TIME DISTRIBUTION From (5), we need to derive si andf si (t). To determine si, the activity of the PUs is observed over time interval [,T], where T is the length of the first N idle/busy cycles of the unionized PU. Ik i,bi k,i k, and B k are the kth idle and busy eriods of the unionized PU and PU i resectively. Therefore, T = I k +B k N k=1 N i = ( Im i +Bi m )+R i, i 1,...,M}, m=1 (A-1) where R i < I 1 N i+1 +B1 N i+1, and in general N i N j N for i,j 1,...,M. Since PU i must end a transmission for every idle/busy cycle, each PU ends N i transmissions during T. An idle eriod started by PU i results if all other PUs are idle when PU i transits from busy to idle state. Defining N si as the number of idle eriods started by PU i, we know that as T, with lim T N si = N i lim T α j, j i N si = N. With these equations, si can be shown to be si = 1 v i M 1 v j. (A-2) We now determine f si (t). From (8), we need to derive φ ij (t). If i = j, then φ ij (t) = φ ii (t) is φ ii (t) = Pr[min(V1 r,v 2 r,...,v i,...,vm r ) = V i] = f Vi (t) [1 F V r j (t)]. j i (A-3) If i j, φ ij (t) is the robability that the residual idle time of PU j is t, and that all other users idle times are greater than t, or φ ij (t) = Pr[min(V1 r,v2 r,...,vj r,...,v i,...,vm) r = Vj r ] = f V r j (t)[1 F Vi (t)] [1 F V r k (t)]. k=1 k i,j (A-4) We have now derived all terms from (5) for the full unionized idle time distribution f V (t). From (5) we can obtain v, and combining with (1), we can obtain n as n = v ( 1 α 1). REFERENCES (A-5) [1] Sectrum olicy task force reort, Federal Communications Commision, Reort, Et docket No , November 22. [2] Facilitating oortunities for flexible, efficient, and reliable sectrum use emloying cognitive radio technologies, notice of roosed rule making and order, Federal Communications Commision, Reort, Et docket No , December 23. [3] Q. Zhao and B. Sadler, A survey of dynamic sectrum access: Signal rocessing, networking, and regulatory olicy, IEEE Signal Processing Magazine, vol. 55, no. 5, , 27. [4] S. Haykin, Cognitive radio: brain-emowered wireless communications, IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, , 25, [5] J. Mitola and J. Maguire, G. Q., Cognitive radio: making software radios more ersonal, Personal Communications, IEEE [see also IEEE Wireless Communications], vol. 6, no. 4, , 1999, [6] Shared sectrum comany: DARPA XG rogram information. [Online]. Available: htt:// [7] F. W. Seelig, A descrition of the august 26 xg demonstrations at fort a.. hill, in Second IEEE International Symosium on Dynamic Sectrum Access Networks, DySPAN, 27, [8] S. Huang, X. Liu, and Z. Ding, Oortunistic sectrum access in cognitive radio networks, IEEE INFOCOM 28, Phoenix, AZ, USA, Aril, 28. [Online]. Available: htt:// senhua/osainfocom8.df

14 JUNG AND LIU: OPPORTUNISTIC SPECTRUM ACCESS IN MULTIPLE PRIMARY USER ENVIRONMENTS UNDER THE PACKET COLLISION CONSTRAINT 14 [9] Q. Zhao, S. Geirhofer, L. Tong, and B. M. Sadler, Otimal dynamic sectrum access via eriodic channel sensing, in Proc. Wireless Communications and Networking Conference (WCNC), 27. [1] T. Shu, S. Cui, and M. Krunz, Medium access control for multi-channel arallel transmission in cognitive radio networks, Proceedings of the IEEE GLOBECOM 26 Conference, San Francisco, CA, Dec., 26. [11] R. Urgaonkar and M. J. Neely, Oortunistic scheduling with reliability guarantees in cognitive radio networks, IEEE INFOCOM 28, Phoenix, AZ, USA, Ar., 28. [12] W. Wang and X. Liu, List-coloring based channel allocation for oensectrum wireless networks, Proceedings of the IEEE Vehicular Tech. Conference, , 25. [13] H. Zheng and C. Peng, Collaboration and fairness in oortunistic sectrum access, Proceedings of the IEEE ICC Conference, 25. [14] L. Cao and H. Zheng, Understanding the ower of distributed coordination for dynamic sectrum management, ACM/Sringer Mobile Networks and Alications (MONET),, vol. 13, [15] L. Yang, L. Cao, and H. Zheng, Physical interference driven dynamic sectrum management, Proceedings of IEEE Symosium on New Frontiers in Dynamic Sectrum Access Networks(DySPAN). [16] Y. S. Y. Thomas Hou and H. D. Sherali, Sectrum sharing for multiho networking with cognitive radios, IEEE Journal on Selected Areas in Communications, vol. 26, no. 1, 28. [17] T. Shu and M. Krunz, Coordinated channel access in cognitive radio networks: A multi-level sectrum oortunity ersective, IEEE INFO- COM 29, Rio De Janeiro, Brazil, Br, 29. [18] D. Xu, E. Jung, and X. Liu, Otimal bandwidth selection in multichannel cognitive radio networks: How much is too much? Third IEEE International Symosium on Dynamic Sectrum Access Networks, DySPAN, 28. [19] D. I. Kim, L. B. Le, and E. Hoossain, Joint rate and ower allocation for cognitive radios in dynamic sectrum access environment, IEEE Transactions on Wireless Communications, vol. 7, no. 12, , 28. [2] Y. Chen, G. Yu, Z. Zhang, H.-H. Chen, and P. Qiu, On cognitive radio networks with oortunistic ower control strategies in fading channels, IEEE Transactions on Wireless Communications, vol. 7, no. 7, , 28. [21] In the matter of establishment of an interference temerature metric to quantify and manage interference and to exand available unlicensed oeration in certain fixed, mobile and satellite frequency bands, Federal Communications Commision/Office of Engineering and Technology, OET Reort, FCC 7-78, May 27. [22] Q. Zhao, L. Tong, A. Swami, and Y. Chen, Decentralized cognitive mac for oortunistic sectrum access in ad hoc networks: A omd framework, IEEE Journal on Selected Areas in Communications, vol. 25, no. 3, , 27. [23] Y. Chen, Q. Zhao, and A. Swami, Joint design and searation rincile for oortunistic sectrum access in the resence of sensing errors, Proc. of IEEE Asilomar Conference on Signals, Systems, and Comuters, November,, 26. [24] T. Javidi, B. Krishnamachari, Q. Zhao, and M. Liu, Otimality of myoic sensing in multi-channel oortunistic access, Proc. of IEEE International Conference on Communications (ICC), May, 28. [25] S. Huang, X. Liu, and Z. Ding, Otimal transmission strategies for dynamic sectrum access in cognitive radio networks, IEEE Transactions on Mobile Communications, vol. 8, no. 12, 29. [26] E. Jung and X. Liu, Oortunistic sectrum access in heterogeneous user environments, Third IEEE International Symosium on Dynamic Sectrum Access Networks, DySPAN, 28. [27] D. Willkomm, S. Machiraju, J. Bolot, and A. Wolisz, Primary users in cellular networks: A large-scale measurement study, Proceedings of IEEE Symosium on New Frontiers in Dynamic Sectrum Access Networks(DySPAN). [28] I. F. Akyildiz, W.-Y. Lee, M. C. Vuran, and S. Mohanty, A survey on sectrum management in cognitive radio networks, IEEE Communications Magazine, Aril 28. [29] Y. Yuan, P. Bahl, R. Chandra, T. Moscibroda, and Y. Wu, Allocating dynamic time-sectrum blocks in cognitive radio networks, in MobiHoc 7: Proceedings of the 8th ACM international symosium on Mobile ad hoc networking and comuting. New York, NY, USA: ACM, 27, [3] S. Boyd and L. Vandenberghe, Convex Otimization, 1st ed. New York, NY, U.S.A.: Cambridge University Press, 24. Eric Jung is currently a Ph.D. candidate in the Electrical and Comuter Engineering Deartment at University of California, Davis. He received his B.S. and M.S. in electrical engineering from the same university in 26 and 29, resectively. His research interests include wireless networking and communications, with an emhasis on dynamic sectrum access. Xin Liu received the PhD degree in electrical engineering from Purdue University in 22. She is currently an associate rofessor in the Comuter Science Deartment at the University of California, Davis. Before joining UC Davis, she was a ostdoctoral research associate in the Coordinated Science Laboratory at the University of Illinois at Urbana- Chamaign. Her research is on wireless communication networks with a focus on resource allocation and dynamic sectrum management. She received the Best Paer of Year Award from the Comuter Networks Journal in 23 for her work on oortunistic scheduling. She received the US National Science Foundation CAREER Award in 25 for her research on Smart-Radio- Technology-Enabled Oortunistic Sectrum Utilization. She received the Outstanding Engineering Junior Faculty Award from the College of Engineering, UC Davis, in 25. She is a member of the IEEE.